The Ohm's law works there in one of its most educational ways.
Joule's heat can be calculated as $$P=UI$$ where U is the voltage drop over the part and I is a current through it.
Ohm's law says $$R=\frac UI.$$
Putting this together we know that high current power source was used. The resistance and current are therefore known and we have enough information to estimate the heating power as $$P=RI^2.$$
The highest resistance is at the contact between the spanner and the clamps and the crossection is lowest there as well, that's why the glowing started there and propagated through whole spanner.
- the higher current, the higher heating power and thus higher temperature
- the higher resistance, the higher heating power. (One need to provide higher voltage to sustain same current)
- metals have higher resistance when heated, therefore the hot parts are heated even more
- The thinner and longer the conductor is, the higher resistance it has, therefore the narrow part is heated more
- The thinner part has smaller weight so its temperature rises even faster,
- Metals usually have higher heat conductivity, so the heat spreads through the spanner effectively increasing the resistance in the "colder" parts.